From 18e03920aa437ad6ee3303d56eab6b569b7fee7f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Mon, 21 Oct 2019 18:19:30 +0200 Subject: [PATCH] some work on the computations --- computations/Untitled.ipynb | 568 +------------------------ computations/stuff_with_sqrtiZ.ipynb | 599 +++++++++++++++++++++++++++ 2 files changed, 602 insertions(+), 565 deletions(-) create mode 100644 computations/stuff_with_sqrtiZ.ipynb diff --git a/computations/Untitled.ipynb b/computations/Untitled.ipynb index 171f561..4b7ac78 100644 --- a/computations/Untitled.ipynb +++ b/computations/Untitled.ipynb @@ -7,571 +7,9 @@ "outputs": [], "source": [ "import sympy\n", - "sympy.init_printing()" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [], - "source": [ - "a, b, c, d = sympy.symbols(\"a b c d\")\n", - "i = sympy.I" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], - "source": [ - "X = sympy.Matrix([[0, 1], [1, 0]])\n", - "Z = sympy.Matrix([[1, 0], [0, -1]])\n", - "\n", - "M = sympy.Matrix([[a, b], [c, d]])" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡0 1⎤\n", - "⎢ ⎥\n", - "⎣1 0⎦" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "X" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡1 0 ⎤\n", - "⎢ ⎥\n", - "⎣0 -1⎦" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "Z" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡a b⎤\n", - "⎢ ⎥\n", - "⎣c d⎦" - ] - }, - "execution_count": 6, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "M" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": {}, - "outputs": [], - "source": [ - "term_X = M*M + i*X\n", - "term_Z = M*M - i*Z" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c & a b + b d + i\\\\a c + c d + i & b c + d^{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ 2 ⎤\n", - "⎢ a + b⋅c a⋅b + b⋅d + ⅈ⎥\n", - "⎢ ⎥\n", - "⎢ 2 ⎥\n", - "⎣a⋅c + c⋅d + ⅈ b⋅c + d ⎦" - ] - }, - "execution_count": 8, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "term_X" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c - i & a b + b d\\\\a c + c d & b c + d^{2} + i\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ 2 ⎤\n", - "⎢a + b⋅c - ⅈ a⋅b + b⋅d ⎥\n", - "⎢ ⎥\n", - "⎢ 2 ⎥\n", - "⎣ a⋅c + c⋅d b⋅c + d + ⅈ⎦" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "term_Z" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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\n", - "text/latex": [ - "$\\displaystyle \\left[ \\left\\{ a : - \\frac{\\sqrt{2}}{2}, \\ b : \\frac{\\sqrt{2} i}{2}, \\ c : \\frac{\\sqrt{2} i}{2}, \\ d : - \\frac{\\sqrt{2}}{2}\\right\\}, \\ \\left\\{ a : \\frac{\\sqrt{2}}{2}, \\ b : - \\frac{\\sqrt{2} i}{2}, \\ c : - \\frac{\\sqrt{2} i}{2}, \\ d : \\frac{\\sqrt{2}}{2}\\right\\}, \\ \\left\\{ a : - \\frac{\\sqrt{2} i}{2}, \\ b : \\frac{\\sqrt{2}}{2}, \\ c : \\frac{\\sqrt{2}}{2}, \\ d : - \\frac{\\sqrt{2} i}{2}\\right\\}, \\ \\left\\{ a : \\frac{\\sqrt{2} i}{2}, \\ b : - \\frac{\\sqrt{2}}{2}, \\ c : - \\frac{\\sqrt{2}}{2}, \\ d : \\frac{\\sqrt{2} i}{2}\\right\\}\\right]$" - ], - "text/plain": [ - "⎡⎧ -√2 √2⋅ⅈ √2⋅ⅈ -√2 ⎫ ⎧ √2 -√2⋅ⅈ -√2⋅ⅈ √2⎫ ⎧\n", - "⎢⎨a: ────, b: ────, c: ────, d: ────⎬, ⎨a: ──, b: ──────, c: ──────, d: ──⎬, ⎨\n", - "⎣⎩ 2 2 2 2 ⎭ ⎩ 2 2 2 2 ⎭ ⎩\n", - "\n", - " -√2⋅ⅈ √2 √2 -√2⋅ⅈ ⎫ ⎧ √2⋅ⅈ -√2 -√2 √2⋅ⅈ⎫⎤\n", - "a: ──────, b: ──, c: ──, d: ──────⎬, ⎨a: ────, b: ────, c: ────, d: ────⎬⎥\n", - " 2 2 2 2 ⎭ ⎩ 2 2 2 2 ⎭⎦" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.solve(term_X)" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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\n", - "text/latex": [ - "$\\displaystyle \\left[ \\left\\{ a : - \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : - \\sqrt{- i}\\right\\}, \\ \\left\\{ a : - \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : \\sqrt{- i}\\right\\}, \\ \\left\\{ a : \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : - \\sqrt{- i}\\right\\}, \\ \\left\\{ a : \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : \\sqrt{- i}\\right\\}\\right]$" - ], - "text/plain": [ - "⎡⎧ ____⎫ ⎧ ____⎫ ⎧ \n", - "⎢⎨a: -√ⅈ, b: 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: -√ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b:\n", - "⎣⎩ ⎭ ⎩ ⎭ ⎩ \n", - "\n", - " ____⎫ ⎧ ____⎫⎤\n", - " 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬⎥\n", - " ⎭ ⎩ ⎭⎦" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.solve(term_Z)" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡-√2 √2⋅ⅈ⎤\n", - "⎢──── ────⎥\n", - "⎢ 2 2 ⎥\n", - "⎢ ⎥\n", - "⎢√2⋅ⅈ -√2 ⎥\n", - "⎢──── ────⎥\n", - "⎣ 2 2 ⎦" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sqrt_miX = sympy.Matrix([[-sympy.sqrt(2)/2, i*sympy.sqrt(2)/2], [i*sympy.sqrt(2)/2, -sympy.sqrt(2)/2]])\n", - "sqrt_miX" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}- \\sqrt{i} & 0\\\\0 & - \\sqrt{- i}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡-√ⅈ 0 ⎤\n", - "⎢ ⎥\n", - "⎢ ____⎥\n", - "⎣ 0 -╲╱ -ⅈ ⎦" - ] - }, - "execution_count": 13, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sqrt_iZ = sympy.Matrix([[-sympy.sqrt(i), 0], [0, -sympy.sqrt(-i)]])\n", - "sqrt_iZ" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}0 & - i\\\\- i & 0\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡0 -ⅈ⎤\n", - "⎢ ⎥\n", - "⎣-ⅈ 0 ⎦" - ] - }, - "execution_count": 14, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sqrt_miX**2" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}i & 0\\\\0 & - i\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ⅈ 0 ⎤\n", - "⎢ ⎥\n", - "⎣0 -ⅈ⎦" - ] - }, - "execution_count": 15, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sqrt_iZ**2" - ] - }, - { - "cell_type": "code", - "execution_count": 16, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}- \\frac{i \\sqrt{- i}}{2} - \\frac{\\sqrt{i} i}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ ____ ____ ⎤\n", - "⎢ ⅈ⋅╲╱ -ⅈ √ⅈ⋅ⅈ √ⅈ ╲╱ -ⅈ ⎥\n", - "⎢- ──────── - ──── - ── + ────── ⎥\n", - "⎢ 2 2 2 2 ⎥\n", - "⎢ ⎥\n", - "⎢ ____ ____⎥\n", - "⎢ √ⅈ ╲╱ -ⅈ √ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ⎥\n", - "⎢ - ── + ────── ──── + ────────⎥\n", - "⎣ 2 2 2 2 ⎦" - ] - }, - "execution_count": 16, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "H_ = sqrt_miX * sqrt_iZ**3 * sqrt_miX\n", - "H_" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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\n", - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{i \\left(\\sqrt{- i} + \\sqrt{i}\\right)}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ ⎛ 5/2⎞ ____ ⎤\n", - "⎢ⅈ⋅⎝-√ⅈ + (-ⅈ) ⎠ √ⅈ ╲╱ -ⅈ ⎥\n", - "⎢───────────────── - ── + ────── ⎥\n", - "⎢ 2 2 2 ⎥\n", - "⎢ ⎥\n", - "⎢ ____ ⎛ ____ ⎞⎥\n", - "⎢ √ⅈ ╲╱ -ⅈ ⅈ⋅⎝╲╱ -ⅈ + √ⅈ⎠⎥\n", - "⎢ - ── + ────── ───────────────⎥\n", - "⎣ 2 2 2 ⎦" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "H_.simplify()\n", - "H_" - ] - }, - { - "cell_type": "code", - "execution_count": 18, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": "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\n", - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡√2 √2 ⎤\n", - "⎢── ── ⎥\n", - "⎢2 2 ⎥\n", - "⎢ ⎥\n", - "⎢√2 -√2 ⎥\n", - "⎢── ────⎥\n", - "⎣2 2 ⎦" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "H = sympy.Matrix([[1, 1], [1, -1]]) / sympy.sqrt(2)\n", - "H" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": {}, - "outputs": [], - "source": [ - "phi = sympy.symbols(\"phi\")" - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAVCAYAAAB7R6/OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARUlEQVQoFWNgYGAoA+K7UGyEhc/QCRR0+f//PwMyBoqBxZmADLxgVAEkeEbDgchwYIEmp5mMjIwgZigQuwBxOhALAfFuAP68FOR8/BSoAAAAAElFTkSuQmCC\n", - "text/latex": [ - "$\\displaystyle \\left[ \\right]$" - ], - "text/plain": [ - "[]" - ] - }, - "execution_count": 20, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.solve(sympy.exp(i*phi)*H_ - H)" - ] - }, - { - "cell_type": "code", - "execution_count": 28, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": "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\n", - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(\\left(- i\\right)^{\\frac{9}{2}} + \\sqrt{i}\\right)}{2} & \\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2}\\\\\\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2} & \\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ ⎛ 9/2 ⎞ 5/2 ⎤\n", - "⎢ⅈ⋅⎝(-ⅈ) + √ⅈ⎠ (-ⅈ) √ⅈ ⎥\n", - "⎢──────────────── ─────── + ── ⎥\n", - "⎢ 2 2 2 ⎥\n", - "⎢ ⎥\n", - "⎢ 5/2 ⎛ 5/2⎞⎥\n", - "⎢ (-ⅈ) √ⅈ ⅈ⋅⎝-√ⅈ + (-ⅈ) ⎠⎥\n", - "⎢ ─────── + ── ─────────────────⎥\n", - "⎣ 2 2 2 ⎦" - ] - }, - "execution_count": 28, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "H__ = H_ * sympy.exp(i * sympy.pi )\n", - "H__.simplify()\n", - "H__" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Some other calculatuions:" - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAAOCAYAAADjXQYbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVQYGX2QARGCQBBF9xgDkAErGMEIWEEb0AEaaAVtYATHCFhBGpzv3+ydDCPuzGd3///HfbAYo81hZh04i6sYlnWCmEQGOdbq18ni/StuZAsh1LQj2IKRqwa6paS03u/eM781CxWnGogHUB3AK03+qLOTXdG7spfBrHXxa56JT8Rr3tVzIKXVK9tEesD8nTsEGW8ecNRefh/knV2YMF0kfgByKG0w9dMLeQAAAABJRU5ErkJggg==\n", - "text/latex": [ - "$\\displaystyle i$" - ], - "text/plain": [ - "ⅈ" - ] - }, - "execution_count": 30, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sympy.exp(i * sympy.pi/2)" - ] - }, - { - "cell_type": "code", - "execution_count": 31, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": "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\n", - "text/latex": [ - "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\\\\\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2} & - \\frac{\\sqrt{- i}}{2} + \\frac{\\sqrt{i}}{2}\\end{matrix}\\right]$" - ], - "text/plain": [ - "⎡ ____ ____⎤\n", - "⎢ √ⅈ ╲╱ -ⅈ √ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ⎥\n", - "⎢ - ── + ────── ──── + ────────⎥\n", - "⎢ 2 2 2 2 ⎥\n", - "⎢ ⎥\n", - "⎢ ____ ____ ⎥\n", - "⎢√ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ╲╱ -ⅈ √ⅈ ⎥\n", - "⎢──── + ──────── - ────── + ── ⎥\n", - "⎣ 2 2 2 2 ⎦" - ] - }, - "execution_count": 31, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "sqrt_miX * sqrt_iZ * sqrt_miX" + "i = sympy.I\n", + "exp = sympy.exp\n", + "M = sympy.Matrix" ] } ], diff --git a/computations/stuff_with_sqrtiZ.ipynb b/computations/stuff_with_sqrtiZ.ipynb new file mode 100644 index 0000000..171f561 --- /dev/null +++ b/computations/stuff_with_sqrtiZ.ipynb @@ -0,0 +1,599 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import sympy\n", + "sympy.init_printing()" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "a, b, c, d = sympy.symbols(\"a b c d\")\n", + "i = sympy.I" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "X = sympy.Matrix([[0, 1], [1, 0]])\n", + "Z = sympy.Matrix([[1, 0], [0, -1]])\n", + "\n", + "M = sympy.Matrix([[a, b], [c, d]])" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 1⎤\n", + "⎢ ⎥\n", + "⎣1 0⎦" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "X" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 0 ⎤\n", + "⎢ ⎥\n", + "⎣0 -1⎦" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡a b⎤\n", + "⎢ ⎥\n", + "⎣c d⎦" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "M" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "term_X = M*M + i*X\n", + "term_Z = M*M - i*Z" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c & a b + b d + i\\\\a c + c d + i & b c + d^{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ 2 ⎤\n", + "⎢ a + b⋅c a⋅b + b⋅d + ⅈ⎥\n", + "⎢ ⎥\n", + "⎢ 2 ⎥\n", + "⎣a⋅c + c⋅d + ⅈ b⋅c + d ⎦" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "term_X" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c - i & a b + b d\\\\a c + c d & b c + d^{2} + i\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ 2 ⎤\n", + "⎢a + b⋅c - ⅈ a⋅b + b⋅d ⎥\n", + "⎢ ⎥\n", + "⎢ 2 ⎥\n", + "⎣ a⋅c + c⋅d b⋅c + d + ⅈ⎦" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "term_Z" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/latex": [ + "$\\displaystyle \\left[ \\left\\{ a : - \\frac{\\sqrt{2}}{2}, \\ b : \\frac{\\sqrt{2} i}{2}, \\ c : \\frac{\\sqrt{2} i}{2}, \\ d : - \\frac{\\sqrt{2}}{2}\\right\\}, \\ \\left\\{ a : \\frac{\\sqrt{2}}{2}, \\ b : - \\frac{\\sqrt{2} i}{2}, \\ c : - \\frac{\\sqrt{2} i}{2}, \\ d : \\frac{\\sqrt{2}}{2}\\right\\}, \\ \\left\\{ a : - \\frac{\\sqrt{2} i}{2}, \\ b : \\frac{\\sqrt{2}}{2}, \\ c : \\frac{\\sqrt{2}}{2}, \\ d : - \\frac{\\sqrt{2} i}{2}\\right\\}, \\ \\left\\{ a : \\frac{\\sqrt{2} i}{2}, \\ b : - \\frac{\\sqrt{2}}{2}, \\ c : - \\frac{\\sqrt{2}}{2}, \\ d : \\frac{\\sqrt{2} i}{2}\\right\\}\\right]$" + ], + "text/plain": [ + "⎡⎧ -√2 √2⋅ⅈ √2⋅ⅈ -√2 ⎫ ⎧ √2 -√2⋅ⅈ -√2⋅ⅈ √2⎫ ⎧\n", + "⎢⎨a: ────, b: ────, c: ────, d: ────⎬, ⎨a: ──, b: ──────, c: ──────, d: ──⎬, ⎨\n", + "⎣⎩ 2 2 2 2 ⎭ ⎩ 2 2 2 2 ⎭ ⎩\n", + "\n", + " -√2⋅ⅈ √2 √2 -√2⋅ⅈ ⎫ ⎧ √2⋅ⅈ -√2 -√2 √2⋅ⅈ⎫⎤\n", + "a: ──────, b: ──, c: ──, d: ──────⎬, ⎨a: ────, b: ────, c: ────, d: ────⎬⎥\n", + " 2 2 2 2 ⎭ ⎩ 2 2 2 2 ⎭⎦" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.solve(term_X)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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YHTSMq2ghcCqugxZPcy1ErMG8jdGC+RjjInN5eu2DY3Ydm/Cd2005h4+H0DL4GZxPjLng4TIfvJ+IHXM9jBYhlio6ZBwurkXMr8V0Ve9FUjg9jV1ih+vsWkYbk73UI+YaVseIDbFCdf5u4f1YDG3eYXTM8KqiZcCuuI6t5X/KNexXe39ptojriWs47u6LTH7GHmkL1cn75bzoG/TeEP39nghDx3nhkxf8ckImf7XAsRvc4NNg56eKrD+vf7VB/0I/5xfKc0cX9T3q8gkoG178MO1+PoUP3LT71eLaMvwFb6LfLxjDWPmRrVPBNZP5d2I8/EL5M6Y4p/D/F/b4xZrOQtuo3ljafO8whnGfCufACXee//2h5bzrnhIT/eaThhcFtoj9u4vGxQX9Qe0qJuNHEu8ZZmhsC2hSL2N4n2qT451rxzE5dehmj/vTWGKK8xSuTlNNxyo6SGON+UAXYjjhzGeGXETpYCiWFXWQrlXRwlox5dDCDLyP4S1zkVKcvHdMGqRjnCPD+lm6UkXLGuXTmY9MOEZpUSgWtdYtiFFFh0jAGjHl0CH6GpovjslYWugY3XeuHUdsQVqWaf3NXamiYzSYyIfYfDrzkQHLKB0LxaLGmp8RooqW1Yoph5Yl8pbQxnJ3lpbgUyfvHTMF6dA0x12cPEb9FkBPH7Ob7q0eASZ30LgB82y1AxptYf6I0/+h8r8m/ITKDTICybY3GL/86CSaPxbMwY0k9nPa+dg7/xnscxOOG0/csOPHr5wbd5N1jOHuLOutN+kWD7s+wlzEIqhgPN/MEz/i8jkqv4ONArZbzLY3lmaLjyP+ZZP/C8cXsHeRswwxTQtrskNzP/jg02NMhlsS7wlAaGwck6vAdjDvc9iGXefaWc6PvtSg6aOKy9ur14Yp70Wtv+Wk5kNxHaRd8917/S59Tb2G/WBOYIwznxYP3QrOBcZG6yANmm0vLM1WcR00v6poYa2YzE6SFobkihguC8YH83Y5R+g1bDp5v5wLfYN1jHMYLjwNXj8ctCzmR3EtM7+91t7SR15jfHA+McaZD/OHU0fhiPHRWmS2vbEwW8W1CHaq6JDlk/gVjclwS9Ih87XZ+xHEEMx7+pxaYNe5dpZzo2+wlmHM9LBA1Ppz+FBcx2jXfPdev3NfMTYqnxjnzEcKlhgbrWOhWJitomt+whq2qmhZrZjMTpKWYY4kHcP4KO5OOYk5wqaT98v50DdIh6z/Z7EbYRQAr42zpaNHuQZA3CDkU1f8zq8ssdqcxO7mKDjs+amY9hA61v0SvM+NAHyk2PEpza9D5s7JVfNheB0kviU4kTMXIRwo1Xe0eGZ5P+zvqxK8zckf+BelY7lzY350r2Ul8mlzHpbja3xUTGuoHLetBO9zowEfo7QsJ1fNh6vUMeYzJ5a5+REz32jxjJAjy0nWfZEYbrjGmAZ4vze0/p994prw2tvxJptPW/EpMD5ql6vwI6HDbIIZKIopFzs6mKcQ70tEFrOORuRqCWwv5izEidFyMVo85MChYyrE24u1keEiRscOn5sY3Arl89Acd+ComBzAHLG5EO9LQBGjZSNydRPbgvkcDcvR4iEvDh1TQe5urpnAm8E69GmggavqjqTzI4hZCnYe72Giv7JM1skkiqmTRGR2IyfvM7sWPd2IXI0GI2JgTk6MlovR4iE9RokpJ28jlk2RIaPkJgacnPkcEUfFFMOq/sfk5H0v0Y7IVV9sc+dzNCxHi4e8GCWm3Nz1XTMl++mJsJLoXs79DQj0n8umw18ppsOn8GoCGJGrR03eaLkYLR7yasSYjrpeln4rN0tE4q5HxFExxXFBo+ojMCJX66P4weJoWI4WD7M0Ykyt+J7VrjbCssLpngybYPwvFkMVxTRUOocOZkSuHjVho+VitHjIqxFjOup6Wfqt3CwRibseEUfFFMcFjaqPwIhcrY/iB4ujYTlaPMzSiDG14ntuu9oIy42o5hMCQkAICAEhIASEgBAQAkJACAgBISAEhIAQ6BIBbYR1mRY5JQSEgBAQAkJACAgBISAEhIAQEAJCQAgIASGQGwFthOVGVPMJASEgBISAEBACQkAICAEhIASEgBAQAkJACHSJQMxG2DtEwv+AqCIEhIAQuFYEpIPXmnnFLQTGQkBaNlY+FY0QuEYEpGPXmHXFLAQSEYjZCLuBzfuJdjVcCAgBIRCMgP0LYm7E80VPyyIdbIm+bAuBAyPQkY4RRWnZgbkk14VASwQ60jLpWEsiyLYQaIhAig7FbIS9QawPYfRBw5hlWggIgetE4HsL+2Xj8KWDjRMg80LgwAj0omOEUFp2YCLJdSHQGIFetEw61pgIMi8EGiIQrUMxG2EvEOgfqK9tB65h3DItBITAtSAAvXmOWFmf4F8RU4NaFulgS/RlWwgcFIHOdIwoSssOyiW5LQRaItCZlknHWpJBtoVAIwRSdWjaCHuMid7P6lNXPHgDysdPH6HyjeifGOPs65pD7UJACAgBXwSgMXwC9Xf0f4b6CBr0i+/YUv2kg6WQ1bxCYEwEetQxIi0tG5NvikoIlEKgRy2TjpXKtuYVAn0iEKpD6P8K9bzXhajuMbK7EI/oCDHhYwz+FvUF5nkbPZEGCgEhIAQcCFDseKtXjZEOOhKnZiEgBM4I9K5jdFRadk6XToSAEHAg0LuWSccciVOzEBgIgVw69H+Nc+igScEWfAAAAABJRU5ErkJggg==\n", + "text/latex": [ + "$\\displaystyle \\left[ \\left\\{ a : - \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : - \\sqrt{- i}\\right\\}, \\ \\left\\{ a : - \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : \\sqrt{- i}\\right\\}, \\ \\left\\{ a : \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : - \\sqrt{- i}\\right\\}, \\ \\left\\{ a : \\sqrt{i}, \\ b : 0, \\ c : 0, \\ d : \\sqrt{- i}\\right\\}\\right]$" + ], + "text/plain": [ + "⎡⎧ ____⎫ ⎧ ____⎫ ⎧ \n", + "⎢⎨a: -√ⅈ, b: 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: -√ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b:\n", + "⎣⎩ ⎭ ⎩ ⎭ ⎩ \n", + "\n", + " ____⎫ ⎧ ____⎫⎤\n", + " 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬⎥\n", + " ⎭ ⎩ ⎭⎦" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.solve(term_Z)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡-√2 √2⋅ⅈ⎤\n", + "⎢──── ────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2⋅ⅈ -√2 ⎥\n", + "⎢──── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt_miX = sympy.Matrix([[-sympy.sqrt(2)/2, i*sympy.sqrt(2)/2], [i*sympy.sqrt(2)/2, -sympy.sqrt(2)/2]])\n", + "sqrt_miX" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- \\sqrt{i} & 0\\\\0 & - \\sqrt{- i}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡-√ⅈ 0 ⎤\n", + "⎢ ⎥\n", + "⎢ ____⎥\n", + "⎣ 0 -╲╱ -ⅈ ⎦" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt_iZ = sympy.Matrix([[-sympy.sqrt(i), 0], [0, -sympy.sqrt(-i)]])\n", + "sqrt_iZ" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & - i\\\\- i & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 -ⅈ⎤\n", + "⎢ ⎥\n", + "⎣-ⅈ 0 ⎦" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt_miX**2" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}i & 0\\\\0 & - i\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ⅈ 0 ⎤\n", + "⎢ ⎥\n", + "⎣0 -ⅈ⎦" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt_iZ**2" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- \\frac{i \\sqrt{- i}}{2} - \\frac{\\sqrt{i} i}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ ____ ____ ⎤\n", + "⎢ ⅈ⋅╲╱ -ⅈ √ⅈ⋅ⅈ √ⅈ ╲╱ -ⅈ ⎥\n", + "⎢- ──────── - ──── - ── + ────── ⎥\n", + "⎢ 2 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ ____ ____⎥\n", + "⎢ √ⅈ ╲╱ -ⅈ √ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ⎥\n", + "⎢ - ── + ────── ──── + ────────⎥\n", + "⎣ 2 2 2 2 ⎦" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H_ = sqrt_miX * sqrt_iZ**3 * sqrt_miX\n", + "H_" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{i \\left(\\sqrt{- i} + \\sqrt{i}\\right)}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ ⎛ 5/2⎞ ____ ⎤\n", + "⎢ⅈ⋅⎝-√ⅈ + (-ⅈ) ⎠ √ⅈ ╲╱ -ⅈ ⎥\n", + "⎢───────────────── - ── + ────── ⎥\n", + "⎢ 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ ____ ⎛ ____ ⎞⎥\n", + "⎢ √ⅈ ╲╱ -ⅈ ⅈ⋅⎝╲╱ -ⅈ + √ⅈ⎠⎥\n", + "⎢ - ── + ────── ───────────────⎥\n", + "⎣ 2 2 2 ⎦" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H_.simplify()\n", + "H_" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2 √2 ⎤\n", + "⎢── ── ⎥\n", + "⎢2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2 -√2 ⎥\n", + "⎢── ────⎥\n", + "⎣2 2 ⎦" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H = sympy.Matrix([[1, 1], [1, -1]]) / sympy.sqrt(2)\n", + "H" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "phi = sympy.symbols(\"phi\")" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAVCAYAAAB7R6/OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARUlEQVQoFWNgYGAoA+K7UGyEhc/QCRR0+f//PwMyBoqBxZmADLxgVAEkeEbDgchwYIEmp5mMjIwgZigQuwBxOhALAfFuAP68FOR8/BSoAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$\\displaystyle \\left[ \\right]$" + ], + "text/plain": [ + "[]" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.solve(sympy.exp(i*phi)*H_ - H)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(\\left(- i\\right)^{\\frac{9}{2}} + \\sqrt{i}\\right)}{2} & \\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2}\\\\\\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2} & \\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ ⎛ 9/2 ⎞ 5/2 ⎤\n", + "⎢ⅈ⋅⎝(-ⅈ) + √ⅈ⎠ (-ⅈ) √ⅈ ⎥\n", + "⎢──────────────── ─────── + ── ⎥\n", + "⎢ 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ 5/2 ⎛ 5/2⎞⎥\n", + "⎢ (-ⅈ) √ⅈ ⅈ⋅⎝-√ⅈ + (-ⅈ) ⎠⎥\n", + "⎢ ─────── + ── ─────────────────⎥\n", + "⎣ 2 2 2 ⎦" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H__ = H_ * sympy.exp(i * sympy.pi )\n", + "H__.simplify()\n", + "H__" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Some other calculatuions:" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAAOCAYAAADjXQYbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVQYGX2QARGCQBBF9xgDkAErGMEIWEEb0AEaaAVtYATHCFhBGpzv3+ydDCPuzGd3///HfbAYo81hZh04i6sYlnWCmEQGOdbq18ni/StuZAsh1LQj2IKRqwa6paS03u/eM781CxWnGogHUB3AK03+qLOTXdG7spfBrHXxa56JT8Rr3tVzIKXVK9tEesD8nTsEGW8ecNRefh/knV2YMF0kfgByKG0w9dMLeQAAAABJRU5ErkJggg==\n", + "text/latex": [ + "$\\displaystyle i$" + ], + "text/plain": [ + "ⅈ" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sympy.exp(i * sympy.pi/2)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\\\\\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2} & - \\frac{\\sqrt{- i}}{2} + \\frac{\\sqrt{i}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ ____ ____⎤\n", + "⎢ √ⅈ ╲╱ -ⅈ √ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ⎥\n", + "⎢ - ── + ────── ──── + ────────⎥\n", + "⎢ 2 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ ____ ____ ⎥\n", + "⎢√ⅈ⋅ⅈ ⅈ⋅╲╱ -ⅈ ╲╱ -ⅈ √ⅈ ⎥\n", + "⎢──── + ──────── - ────── + ── ⎥\n", + "⎣ 2 2 2 2 ⎦" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt_miX * sqrt_iZ * sqrt_miX" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}